Capacity expansion

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This is an augmentation to any mathematical program that has constraints representing a capacity limit. Suppose LaTeX: g(x) \le b represents the limitation of using more than LaTeX: b units of capacity, where LaTeX: g(x) is the actual amount used for policy LaTeX: x. Then, this becomes a capacity expansion model by replacing the constraint with

LaTeX: 
g(x) - vC \le b \;\mbox{ and }\; 0 \le v \le 1.
Here LaTeX: C is the maximum amount of new capacity, with LaTeX: vC the portion brought online by choice of LaTeX: v. This is also reflected in the objective function with an added cost term, say LaTeX: F(v), such that LaTeX: F(0)=0 and LaTeX: F increasing on LaTeX: [0, 1]. If continuous use of the capacity is not possible (i.e., one builds a new plant or not), the model further requires LaTeX: v to be binary-valued (0 or 1), and LaTeX: F(1) is called the fixed charge.


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